Plane Geometry: Angles, Triangles and Polygons

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your NECO exam.

Angle Types:

• Acute: $0° < \theta < 90°$
• Right: $\theta = 90°$
• Obtuse: $90° < \theta < 180°$
• Straight: $\theta = 180°$
• Reflex: $180° < \theta < 360°$
• Complementary angles: Sum to $90°$
• Supplementary angles: Sum to $180°$

Angle Properties:

• Vertically opposite angles are equal
• Angles on a straight line sum to $180°$
• Angles around a point sum to $360°$
• Corresponding angles are equal (parallel lines)
• Alternate angles are equal (parallel lines)

Triangle Types:

• Equilateral: All sides equal, all angles $60°$
• Isosceles: Two sides equal, two angles equal (base angles)
• Scalene: All sides different
• Right-angled: One angle $90°$

Triangle Geometry:

• Angle sum property: Sum of interior angles of any triangle = $180°$
• Exterior angle theorem: An exterior angle of a triangle equals the sum of the two opposite interior angles

Polygon Angle Sums:

• Sum of interior angles of an $n$-sided polygon = $(n - 2) \times 180°$
• Each interior angle of a regular $n$-gon = $\dfrac{(n-2) \times 180°}{n}$
• Sum of exterior angles of any polygon = $360°$

⚡ NECO Tip: In an isosceles triangle, the altitude from the apex bisects the base and the apex angle. This is very useful in NECO geometry problems — look for the line of symmetry.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for NECO Mathematics students with a few days to months.

Congruent Triangles (SSS, SAS, ASA, AAS, RHS):

Two triangles are congruent if:

1. SSS: All three sides equal
2. SAS: Two sides and the included angle equal
3. ASA: Two angles and the included side equal
4. AAS: Two angles and one side equal
5. RHS (Right angle-hypotenuse-side): Right-angled triangles with hypotenuse and one side equal

Similar Triangles: Two triangles are similar if their corresponding angles are equal (AAA). Corresponding sides are in the same ratio.

• Ratio of areas of similar triangles = (ratio of corresponding sides)²

Area of Triangle: $$A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}ab\sin C = \sqrt{s(s-a)(s-b)(s-c)}$$ where $s = \frac{a+b+c}{2}$ (semi-perimeter, Heron’s formula)

Pythagoras’ Theorem: In a right-angled triangle with hypotenuse $c$: $$c^2 = a^2 + b^2$$ If $c^2 = a^2 + b^2$: right-angled If $c^2 < a^2 + b^2$: acute-angled If $c^2 > a^2 + b^2$: obtuse-angled

Circle Theorems:

• Angle in a semicircle = $90°$
• Angles in the same segment are equal
• The angle at the centre is twice the angle at the circumference subtended by the same arc
• The opposite angles of a cyclic quadrilateral sum to $180°$
• Tangent to a circle is perpendicular to the radius at the point of contact

Regular Polygons:

Polygon	Sides	Interior Angle	Exterior Angle
Triangle	3	$60°$	$120°$
Square	4	$90°$	$90°$
Pentagon	5	$108°$	$72°$
Hexagon	6	$120°$	$60°$

⚡ NECO Common Mistakes:

• Confusing “angles in the same segment” with “angles in alternate segments”
• Forgetting that the exterior angle of a triangle equals the sum of the two opposite interior angles (not the adjacent one)
• In Pythagoras’ theorem, misidentifying the hypotenuse (it’s always opposite the right angle)
• Mixing up similar and congruent triangle criteria

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for NECO and JAMB Mathematics preparation.

Coordinate Geometry:

Distance between two points $(x_1, y_1)$ and $(x_2, y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Midpoint of a line segment: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Gradient of a line: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta$$

Two lines are parallel if $m_1 = m_2$. Two lines are perpendicular if $m_1 \times m_2 = -1$.

Equation of a straight line: $$y - y_1 = m(x - x_1) \quad \text{(point-slope form)}$$ $$y = mx + c \quad \text{(slope-intercept form)}$$ $$\frac{x}{a} + \frac{y}{b} = 1 \quad \text{(intercept form)}$$

Angle Between Two Lines: $$\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$$

Area of a Triangle from Coordinates: $$A = \frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|$$

Polygons — Detailed:

For a regular $n$-gon:

• Interior angle $= \frac{(n-2) \times 180°}{n}$
• Exterior angle $= \frac{360°}{n}$
• Each interior angle $+ $ each exterior angle $= 180°$

The Exterior Angle Theorem for Polygons: Each exterior angle of a regular $n$-gon = $\frac{360°}{n}$. This is also the angle by which the polygon turns at each vertex.

Concurrency in Triangles:

• Centroid: Intersection of medians (each median connects a vertex to the midpoint of the opposite side). Divides each median in 2:1 ratio (vertex to centroid is twice centroid to midpoint).
• Circumcentre: Intersection of perpendicular bisectors of sides. Centre of the circumscribed circle.
• Incentre: Intersection of angle bisectors. Centre of the inscribed circle.
• Orthocentre: Intersection of altitudes.

Geometric Constructions (NECO Practical):

• Construct a perpendicular bisector of a line segment
• Construct an angle bisector
• Construct a triangle given SSS
• Construct a tangent to a circle from an external point

NECO/JAMB Patterns:

• NECO frequently asks: prove triangle congruence and similarity; calculate interior and exterior angles of polygons; apply Pythagoras’ theorem; solve geometry problems using circle theorems; find areas of triangles and polygons using formulas

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